Case study :
A teacher asked to Rohan to draw a triangle with following condition: The longest side of the triangle is 7 cm less than twice the shortest side and third side is 7 cm shorter than longest side. The perimeter of the triangle is atleast 84 cm.

Based on the above information, form a linear inequation and answer the following questions :

(i) What is the minimum length of the shortest side ?

(ii) What is the minimum length of the longest side ?

(iii) Identify the type of triangle that Rohan has drawn along with the length possible sides he got.

(iv) What is the least area of the triangle drawn ?

Let the shortest side of triangle be x cm.

Given,

Longest side of triangle = 2x - 7

Third side of triangle = (2x - 7) - 7 = 2x - 14

(i) Given,

Perimeter of triangle is atleast 84 cm.

⇒ x + (2x - 7) + (2x - 14) ≥ 84

⇒ 5x - 21 ≥ 84

⇒ 5x ≥ 84 + 21

⇒ 5x ≥ 105

⇒ x ≥ $\dfrac{105}{5}$

⇒ x ≥ 21.

Hence, minimum length of the shortest side of triangle = 21 cm.

(ii) Length of the longest side of triangle = 2x - 7

Minimum length of longest side will be when x = 21, substituting value we get :

⇒ 2(21) - 7

⇒ 42 - 7

⇒ 35 cm.

Hence, minimum length of the longest side of triangle = 35 cm.

(iii) Minimum length of third side :

Third side of triangle = 2x - 14

= 2(21) - 14

= 42 - 14

= 28 cm.

Three sides of triangle = 21 cm, 28 cm and 35 cm.

⇒ 21² + 28² = 441 + 784 = 1225

⇒ 35² = 1225.

Thus, we can say that 21² + 28² = 35².

Thus, it is a right angled triangle with hypotenuse 35 cm and other two sides are 21 cm and 28 cm.

Hence, the triangle is right-angled triangle.

(iv) By formula,

Area of right angled triangle = $\dfrac{1}{2}$ × product of sides containing right angle

Least area of the triangle = $\dfrac{1}{2}$ × 21 × 28

= 294 cm².

Hence, the least area of the triangle = 294 cm².